Role of particle-size distributions onmillimetre-wave propagation in sand/duststorms

A.S. Ahmed, MSc

Indexing term: Radiowave propagation (millimetre wave)

Abstract: Particle size distributions of sand anddust are surveyed. Various assumptions for thetheoretical prediction of the effect of particles onMMW propagation are discussed and justified. Ageneral formula of the propagation parameters,for any particle shape and any distribution, isdeveloped. It is found that the MMW propaga-tion depends strongly on the probability densityfunction of particle size distribution. A simplelinear relationship is shown between 'a, $' and an'effective radius' of the storm. Attenuation andphase shift due to spherical and ellipsoidal par-ticles are calculated for four distributions, namelypower law, exponential, lognormal and normal.Results are presented and the role of the distribu-tions is discussed.

1 Introduction

One of the parameters on which the propagation of milli-metric wave (MMW) through randomly distributed par-ticles depends, is the particle size distribution. For rain,this distribution is well established and documented [1].For sand and dust, the distributions reported (in theliterature) vary considerably.

The assumed equisized distribution [2] is unrealistic,since various ranges of sizes may exist during a storm,and the measured exponential distribution [3, 4] dependsonly on one sample. The only normal distribution mea-sured [5] is based on a limited number of samples. Theproblem becomes complicated since some authors haveclaimed that the distribution is wind speed dependent[6-8] while others reported that the distribution is windspeed independent [9, 10]. Moreover, the exponent of thepower law distribution, (size)"v, differs from location tolocation and is time dependent [8]. More than one dis-tribution may exist during a storm [11]. Although thesize distribution of aerosols generated by wind erosion ofsoil is a function of the parent soil, it is not identical tothe surface soil distribution [12]. However, the measure-ment of size distribution of aerosols [13] and of bulk soil[6] showed identical power law distribution. Measure-ments of soil-derived aerosols three metres above the soilsurface suggested that the volume size distribution over0.02 to 100 /zm range is three overlapping lognormal. Thefit is reasonable theoretically because soil derived aero-sols are produced by sandblasting process which do

Paper 5060H (Ell), first received 19th May and in revised form 3rdSeptember 1986The author is with the Research Centre, College of Engineering, KingSaud University, PO Box 800, Riyadh 11421, Saudi Arabia

result lognormal distribution [14]. Single-mode lognor-mal distributions [11, 15] and bimodal lognormal dis-tribution [10, 14, 16] have been also measured. Table 1summarises the measured size distributions of sand anddust particles and their different parameters.

The aim of this paper is (i) to present a generalformula for the parameters of propagation of MMW induststorms suitable for any particle shape and size dis-tribution, (ii) to compare and discuss the effect of variousparticle size distributions on MMW propagation and (iii)to find how important is this effect.

2 General formula for the attenuation and phaseshift of MMW in duststorm

2.1 AssumptionsIn general, problems involving randomly distributed par-ticles (suspended in a storm) are complicated and analyti-cal solutions do not exist [19]. Certain simplifyingassumptions have to be employed [20-22]. Theseassumptions are examined in the following, (i) The par-ticles are randomly distributed, infinite in number, identi-cally oriented and homogeneous. This implies that thefield at any point in space is the sum of the incident field(in the absence of any particle) and the scattered fieldsfrom all the particles. For a homogeneous particle, thepermittivity e is not a function of position inside the par-ticle, (ii) The particle suspension in the atmosphere is aweak fluctuation region. This means that random fluc-tuations of the field (incoherent field) are much less thanthe average field (coherent field). This assumption is justi-fied when the scattering is much smaller than the absorp-tion or when the receiving beam width is small (which isnormally the case at millimetric wave), (iii) Single scat-tering approximation applies. Consequently, the fieldincident on each particle is assumed to be equal to thefree-space (external) field, which is scattered only once bya few particles. This situation occurs when the propaga-tion frequency is relatively low or the medium is tenuous,i.e. the particles concentration is < 1 % by volume [23] orthe mutual distance between two adjacent particles ismore than three times the particle size [20]. This is justi-fied even for a severe storm, e.g. particles number concen-tration of the order 108 m~3 [24] and maximum particlesize of 0.1 mm. (iv) The far-field approximation of thescattered field is valid. This implies that the particles arein the far zone of each other which is practically the caseduring dust storms. The scattered field is given by [21]

(1)

where S is the scattering amplitude function 0 and i areunit vectors in the direction of scattering and propaga-tion, respectively, k = 2n/A, r is the distance vector, (v)

IEE PROCEEDINGS, Vol. 134, Pt. H, No. 1, FEBRUARY 1987 55

Table 1 : Measured parameters of size distribution of sand and dust particles

Distributionfunction

Exponential

Normal

Probability density functionp{a), (/;m-1)

/y-1 exp (-a///)

Rangeradiusa, fjm

0.05 -v. 501 -^250

=$10

Meanfj, //m

1415

5.7

SD

1415

2.5

Remarks

One sampleOne sample; H = 3 m4 < l / o < 1 0 k m

15 measurements

References

[3][4]

[5]

Log-normal"

expr (Ina-m)2]L" 2s2 J

0.1 -^100

0.6-v-120

0.6 -v 75

10-V-100

<100

<12

^10

Vg = 1 °g =

// = 3, 50 —

// = 2, 20 —

^o = 21 aff =

2.1 aff <

— —

2 —

2 Single mode; /•/ = 1.5 m

bimodal, wind speeddependent, H = 1 mbimodal, wind speedindependent

1.7 Trimodal,Mode B, l/0 = 8 -v 60 km,H = 2, 6 m

: 2 Trimodal, 329 measurements

11 measurements,independent ofheight

[15]

[10]

[10]

[14]

[17]

[13]

[9]

Power-law**

<101 0 < a>40<20

< 4 02.63.76.51.38 -^2.55

———0.33 •x.10.6

Size dependent

Time and wind speeddependent, WS = 10 ms"1

[18]

[8]

H: Height, m; Vo: optical visibility, km, WS: wind speed, ms~\ SD: standard deviation* m = In fjg, s = In og; jjg and og are geometric parameters

fj = exp {m+s2/2), cr = / / (exps 2 -1 ) 1 / 2

fjgn: Vgm; /jgv: geometric mean of number, mass, volume distributionIn figv (or In fjgm) = In fjgn + 3.0 In2 ag

** C: constant, cm"3

Approximating 5. This approximation is requiredbecause it is difficult to find a solution for the exact inte-gral expression of the scattering amplitude function,which depends on of the local field inside the particle andits permittivity. The local field is generally unknown andthe difficulty is overcome by certain approximations e.g.Rayleigh, Born or Wentzel-Kramers-Brillouin (WKB)[21]. The Rayleigh approximation is valid when theexternal field is homogeneous and continuous, whichoccurs if the particle radius is a ^ 0.05 X (or ka -4 1). Thesecond condition is that the incident field penetrates sofast inside the particle that the static polarisation is estab-lished in a short time compared to the wave period, i.e.k\n\a <̂ 1, where n is the particle refractive index. Bornand WKB approximations are not suitable for the case ofdust particles because of the relatively high value of e (ofthe order of 4.0 —j 1.3) especially at high moisture.

2.2 FormulationAssuming maximum size of suspended particles of theorder of 0.1 mm and a typical refractive indexn = 1.5 — ;0.005 [15], the values of ka and k \ n \ a even at100 GHz are of the order of 0.2 and 0.3, respectively. Thiswould suggest the validity of Rayleigh approximation fordust particles up to 0.1 mm radius and 100 GHz. Twoapproaches may be used to define the forward scatteringamplitude function; the classical molecular optics [20] orthe integral representation method [21]. Under the Ray-leigh approximation, the two approaches lead to thesame result, which in the forward direction (i, i) is given

by

(2)

where p is the complex polarisability of the dipoles of theparticles.

For homogenous identically oriented and equialignedrandomly distributed particles, having particle-sizenumber distribution n(a) and employing single scatteringapproximation, the coherent field inside a slab of par-ticles satisfies the Foldy-Twersky integral equation [21,25]. The solution of this integral equation for an incidentfield e>** is [21]

where the propagation factor K is

K = k + ^ S(i,i)x n{a) dak Jo

(3)

(4)

This shows that an external field which satisfies the waveequation (V2 + k2)Et{r) = 0 in free space, will propagatethrough a randomly distributed particles (tenuousmedium) with a modified propagation constant K givenby eqn. 4. In SI units the attenuation coefficient (a) andthe phase shift coefficient (<f>) are

a = Im <p = Re (5)

which are dependent on the shape of the particle, as willbe shown later.

Spheres: For spherical particles, the complex polarisabil-ity ps is related to the complex permittivity e and the

56 IEE PROCEEDINGS, Vol. 134, Pt. H, No. 1, FEBRUARY 1987

volume of the particle v by Mosotti-Lorentz-Lorenzequation [20, 26]

where

e + 2

(6)

(7)

Considering typical values of e and ka, even up to100 GHz, the forward scattering amplitude functionreduces to

i, i) ~ k2Ga3 (8)

Putting n(a) = Np(a), where N is the number of particlesper unit volume of air (m~3) and p(a) is the probabilitydensity function (m"1), then we can write the attenuationcoefficient for spheres in terms of the measurable quan-tity N, F (GHz), a (m) as

as = (1.143 x 106)[NFG"] a3p(a) da dB/km (9)Jo

In terms of a more convenient storm parameter, namelythe optical visibility Vo, use is made of the optical attenu-ation a0 against Vo relation [2]

Vo = 15/a0 km (10)

where a0 is given by [20]

a0 = 27iJV \a2p(a) da (4.343 x 103) dB/km (11)Jo

Eliminating a0 and substituting, we write

N = 0.551 x 10~3 Vo | a2p(a) da m " 3 (12)

dB/km (13)Jo-a2p(a) da-]

Similarly, it can be shown that

r \[[«>a2 (a)da\ d e g / k m

(14)

Ellipsoids: Most of the small particles, suspended in theatmosphere, can be considered spherical, whereas largerparticles may be ellipsoid. If an ellipsoid is oriented suchthat the field is applied along one of the axes (i = 1, 2, 3)the polarisability p( is given by [20]

(15)

where v is the volume of the ellipsoids, v = (4n/3)abc

/,• are three factors such that lx + l2 + /3 = 1 and/ 1 : / 2 : / 3 = a - 1 : f c - 1 : c - 1 .

A duststorm is assumed to contain identically shapedellipsoidal and equialigned particles with the same orien-tation with respect to the propagation vector. The firstassumption implies that there is no strong dependence ofshape on size [27]. The second assumption is justifiablesince the vast majority of the particles are aligned withtheir longest axes horizontal and their shortest axes verti-cal [27, 28]. The orientation of the particles is considered

to be equal to 90° for horizontal path. The projection ofthe particle into a plane containing the field vectors is anellipse, with i = 1 and i — 2 are the major and minoraxes, respectively. Following a similar procedures to thespherical case, it can be deduced that

a, = (0.381)(106)(JVFiA;') abcp(s) ds dB/km (17)Jo

a0 = (8.686)(103)(nN) PJo

acp(s) ds dB/km (18)

where s denotes the parameters a, b and c. For a size-independent average shape <b/a>, the distribution of theparticle size is a function of the parameter a, consequent-

(19)lfta2p(a)da-]

dB/km

(20)

Eqns. 13, 14, 19 and 20 indicate that, for any particleshape and for any particle size distribution, the propaga-tion of MMW in a duststorm can be represented by asimple and general formula of the linear form

a = v\ae dB/km

<j> = Zae deg/kg

where

(21)

ae =e 7 ,$%a2p(a)da

urn

The 'effective radius' ae depends largely on the particlesize distribution and the size range. The factors r\ and E,are a function of frequency F, particle permittivity e andoptical visibility Vo as shown in eqns. 19 and 20. It isevident that rj and ^ are constants for certain F, e and Vo

and therefore the propagation parameters in a duststormare directly proportional to the 'effective radius', which,in turn, is strongly dependent on the particle size dis-tribution.

3 Calculations and discussions

Generally the probability density function PDF of dustparticle size distribution are best described in practice bya proper fit from observed sample data. It is possible tocompare the effect of different distributions by twoapproaches: (i) all the distributions have the same PDFbut their parameters are changed, (ii) each distributionhas a different PDF and a relation between the param-eters is known. Another problem is that a meaningfulcomparison of data obtained by different investigators isoften difficult, since the data is presented in differentforms [29]. To overcome these problems and to establisha basis of comparison, the parameters are considered tovary around the reported data. The PDF are normalisedto satisfy the condition J p(a) da = 1 in the range10 /im < a < 100 /mi and the integration is zero else-where. This range of radius (mode B) is considered

1EE PROCEEDINGS, Vol. 134, Pt. H, No. 1, FEBRUARY 1987 57

because the effect of smaller sizes and larger sizes can beneglected; the first have smaller ka and the second havehigh fall velocity. The PDF considered are: power law(P), negative exponential (E), normal (N) and lognormal(LN).

The calculated values of the effective radius ae for afamily of normal and lognormal functions are shown inFig. 1. The effective radius is plotted against the average

80

70

60

i 5 0

0*40

30

20

10

cr=

10 20 30 40 50<a>,pm

60 70 80

Fig. 1 Effect of lognormal and normal distributions on the effectiveradius aefor standard deviation a = 5 to a = 20

lognormalnormal

radius <a> for a range of standard deviation a valuesfrom 5 to 20. It is clear from Fig. 1 that, for the samePDF, ae increases with the increase of a a which corre-sponds to a flat function. The increase becomes less pro-nounced at larger values of <a>. For the LN function, afour fold increase in a results in a five fold increase in ae

at <a> = 24 pm, while the increase at <a> = 66 fim isinsignificant. Fig. 2 demonstrates the strong dependenceof ae and the propagation parameters on the PDF of theparticle size distribution. For the expected variation of<fl> and CT, the power law produces the highest propaga-tion parameters values while the normal produces thelowest value (at a = 5). Between these bounds, the rela-tive values of the other functions depend on <a> and a.Also maximum deviation occurs for both distributions.The deviation is more considerable for smaller values of

; approximately 3.6 fold and 1.8 fold for <a> = 26 yum

and 40 fim, respectively. The assumption of equisized dis-tribution generally underestimates the effect of a dust-

80r

10 20 30 40 50 60 70 80

Fig. 2 Effect of different distribution functions on the effective radiusae and the attenuation coefficient asfor spheresF = 37 GHz; e = 4.0 — yl.3; Vo = 0.1 km

normallognormalexponentialpower lawequisized

storm, except for a certain range of N and LNdistributions.

In practical terms, illustrative values of attenuation a,phase shift </>, average attenuation a, differential attenu-ation Aa = cty — a2 and differential phase shift A$ = $!— (j>2 are presented in Table 2 for different distributions

and average radii <a> = 40 yum and 26 fim. The calcu-lation considers spherical and ellipsoidal particles, havingtypical e = 4.0 - ; 1 . 3 at F = 37 GHz and 10% moisturecontent [30], /t = 0.2, /? = 0.5, (b/a) = 0.709 [27, 28]. Arelatively severe storm is assumed (Vo = 0.1 km) along a1 km path. From this example, it is seen that the varia-tion in estimated attenuation would be from 9 to 16 dBfor <a> = 40 ^m on a 10 km path, assuming the completepath to be filled with the duststorm. This will put a limiton the hop distance in a location where relatively humidand severe duststorms occur frequently. However, this isnot always the case, since, for example, the Sudanese andAmerican Haboob are humid [31] while the Khamasinstorm is relatively dry [8]. It is evident also that if thesizes of the ellipsoidal particle are getting smaller, thevalues of Aa and A0 decrease, and consequently the XPDalso decreases. This means that the shape of the particlesdoes not have much effect on depolarisation for smallparticles relative to the wavelength.

Table 2: Illustrative parameters for various distribution functions

Particleshape

Spheres

Ellipsoids

Propagationparameters

as, dB0s- deg

0s- deg

a, dBAa, dBA0, deg

a, dBAa, dBA0, deg

%

40

26

40

26

Normal

C7=5

0.9230.51

0.3712.37

0.720.63

18.03

0.290.257.31

a = 20

1.2942.96

0.8329.95

1.010.88

25.38

0.710.62

17.70

Distribution function

Lognormal

CT=5

1.033.72

0.5819.27

0.800.69

19.93

0.460.40

11.39

a = 20

1.3344.16

1.1638.54

1.040.91

26.10

0.910.79

22.78

Exponential

cr=<a>

1.5551.55

1.1136.94

1.221.06

30.46

0.870.76

21.83

Power law

a

1.6153.80

1.3444.57

1.271.11

31.79

1.050.92

26.33

Equisized

0.9632.12

0.6320.88

0.760.66

18.98

0.490.43

12.34

Path 1 km, Vo = 0.1 km, e = 4.0-/1.3, moisture content 10%, F = 37 GHz, / ^ O Z /2 = 0.5,= 0.709

58 IEE PROCEEDINGS, Vol. 134, Pt. H, No. 1, FEBRUARY 1987

4 Conclusions

Particle size distributions in sand/duststorms vary con-siderably and may be time and height dependent. Theo-retical prediction of MMW propagation intosand/duststorms only employed equisized or exponentialdistribution. To investigate the role of different possibledistributions, a general formula for the propagation intostorms, suitable for any particle shape and any form ofdistribution, is developed. An 'effective radius' is definedae, and a simple linear formula for the propagationparameters is found; (a, 4>) — (rj, £)ae where rj and t, areconstants. The values of ae are calculated for four particlesize distribution functions, namely power law, exponen-tial, lognormal and normal. The results lead to the con-clusion that the form of the size distribution functionaffects significantly the MMW propagation into sand/duststorms. Maximum values and deviation of the propa-gation parameters are produced by the power lawfunction. The equisized assumption underestimates theprediction in general. The relative values for various dis-tribution are considerable, but depend on the averageand the standard deviation of the function. For each spe-cific condition, measurement and proper fitting of thePDF are important if an accurate prediction is to beattained.

5 Acknowledgments

The author would like to thank the College of Engineer-ing, Research Centre, King Saud University for theirencouragement; and to acknowledge the helpful dis-cussions with Professor P.A. Mathewes, The Universityof Leeds, U.K.

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